Relative class numbers inside the $p$th cyclotomic field

For a prime number p ≡ 3 mod 4, we write p = 2 n (cid:2) f + 1 for some power (cid:2) f of an odd prime number (cid:2) and an odd integer n with (cid:2) (cid:2) n . For 0 ≤ t ≤ f , let K t be the imaginary subfield of Q ( ζ p ) of degree 2 (cid:2) t and let h − t be the relative class number of K t . We show that for n = 1 (resp. n ≥ 3), a prime number r does not divide the ratio h − t / h − t − 1 when r is a primitive root modulo (cid:2) 2 and r ≥ (cid:2) f − t − 1 (resp. r ≥ ( n − 2) (cid:2) f − t + 1). In particular, for n = 1 or 3, the ratio h − f / h − f − 1 at the top is not divisible by r whenever r is a primitive root modulo (cid:2) 2 . Further, we show that the (cid:2) -part of h − t / h − t − 1 stabilizes for “large” t under some assumption.